There are a lot of ways to catch or obtain one . . . Just like the following :

[1] The FARM! go to the cat that always rests near a triangular thing, tlak to it and then choose "Black Bug Perfume" and be sure to use some items like: "Honey" and "Hypnotic Pollen" with these items you can have a higher chance of having/catching a "Carpenter Bug" . . .

[2] The QUESTS! Well, just when you can see several lights wanders around at certain spots, these are the spots where you can catch different kinds of bugs. When you are at "High Rank Quests" you'll have better catches of bugs

[3] The QUEST REWARDS! Hmmmmm, these are very rare quests rewards they can be only obtained on "Gathering Quests".